Cubic Interval-Valued Intuitionistic Fuzzy Sets and Their Application in BCK/BCI-Algebras

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1 axioms Article Cubic Interval-Valued Intuitionistic Fuzzy Sets Their Application in BCK/BCI-Algebras Young Bae Jun 1, Seok-Zun Song 2, * ID Seon Jeong Kim 3 1 Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea; skywine@gmail.com 2 Department of Mathematics, Jeju National University, Jeju 63243, Korea 3 Department of Mathematics, Natural Science of College, Gyeongsang National University, Jinju 52828, Korea; skim@gnu.ac.kr * Correspondence: szsong@jejunu.ac.kr; Tel.: Received: 28 December 2017; Accepted: 15 January 2018; Published: 23 January 2018 Abstract: As a new extension of a cubic set, the notion of a cubic interval-valued intuitionistic fuzzy set is introduced, its application in BCK/BCI-algebra is considered. The notions of α-internal, β-internal, α-external β-external cubic IVIF set are introduced, the P-union, P-intersection, R-union R-intersection of α-internal α-external cubic IVIF sets are discussed. The concepts of cubic IVIF subalgebra ideal in BCK/BCI-algebra are introduced, related properties are investigated. Relations between cubic IVIF subalgebra cubic IVIF ideal are considered, characterizations of cubic IVIF subalgebra cubic IVIF ideal are discussed. Keywords: α-internal; β-internal; α-external; β-external cubic interval-valued intuitionistic fuzzy set; cubic interval-valued intuitionistic fuzzy subalgebra; cubic interval-valued intuitionistic fuzzy ideal JEL Classification: MSC; 06F35; 03B60; 03B52 1. Introduction In 2012, Jun et al. [1] introduced cubic sets, then this notion is applied to several algebraic structures (see [2 11]). The aim of this paper is to introduce the notion of a cubic IVIF set which is an extended concept of a cubic set. We introduce cubic interval-valued intuitionistic fuzzy set which is an extension of cubic set, apply it to BCK/BCI-algebra. We investigate P-union, P-intersection, R-union R-intersection of α-internal α-external cubic IVIF sets. We define cubic IVIF subalgebra ideal in BCK/BCI-algebra, investigate related properties. We consider relations between cubic IVIF subalgebra cubic IVIF ideal. We discuss characterizations of cubic IVIF subalgebra cubic IVIF ideal. 2. Preliminaries A fuzzy set in a set X is defined to be a function µ : X [0, 1]. For I = [0, 1], denote by I X the collection of all fuzzy sets in a set X. Define a relation on I X as follows: ( µ, λ I X ) (µ λ ( x X)(µ(x) λ(x))). (1) The complement of µ I X, denoted by µ c, is defined by ( x X) (µ c (x) = 1 µ(x)). (2) Axioms 2018, 7, 7; doi: /axioms

2 Axioms 2018, 7, 7 2 of 17 For a family {µ i i Λ} of fuzzy sets in X, we define the join ( ) meet ( ) operations as follows: µ i : X [0, 1], x sup{µ i (x) i Λ}, µ i : X [0, 1] x inf{µ i (x) i Λ}. An interval-valued intuitionistic fuzzy set (briefly, IVIF set) A over a set X (see [12]) is an object having the form à = { x, α[ã](x), β[ã](x) x X} where α[ã](x) [0, 1] β[ã](x) [0, 1] are intervals for every x X, Especially, if sup α[ã](x) + sup β[ã](x) 1 sup α[ã](x) = inf α[ã](x) sup β[ã](x) = inf β[ã](x), (3) then the IVIF set à is reduced to an intuitionistic fuzzy set (see [13]). Given two closed subintervals D 1 = [D 1, D+ 1 ] D 2 = [D 2, D+ 2 ] of [0, 1], we define the order as follows: D 1 D 2 D 1 D 2 D+ 1 D+ 2. We also define the refined minimum (briefly, rmin) refined maximum (briefly, rmax) as follows: rmin{d 1, D 2 } = [ min{d 1, D 2 }, min{d+ 1, D+ 2 }], rmax{d 1, D 2 } = [ max{d 1, D 2 }, max{d+ 1, D+ 2 }]. For a family {D i = [D i, D + i ] i Λ} of closed subintervals of [0, 1], we define rinf (refined infimum) rsup (refined supermum) as follows: [ rinf D i = inf D i, inf D+ i ] rsupd i = [ sup In this paper we use the interval-valued intuitionistic fuzzy set à = { x, α[ã](x), β[ã](x) x X} D i, sup D + i over X in which α[ã](x) β[ã](x) are closed subintervals of [0, 1] for all x X, that is, α[ã] : X D[0, 1] β[ã] : X D[0, 1] are interval-valued fuzzy (briefly, IVF) sets in X where D[0, 1] is the set of all closed subintervals of [0, 1]. Also, we use the notations α[ã] (x) α[ã] + (x) to mean the left end point the right end point of the interval α[ã](x), respectively, so we have α[ã](x) = [α[ã] (x), α[ã] + (x)]. The interval-valued intuitionistic fuzzy set à = { x, α[ã](x), β[ã](x) x X} over X is simply denoted by Ã(x) = (α[ã](x), β[ã](x)) for x X or à = (α[ã], β[ã]). An algebra (X;, 0) of type (2, 0) is called a BCI-algebra if it satisfies the following axioms: ].

3 Axioms 2018, 7, 7 3 of 17 (I) ( x, y, z X) (((x y) (x z)) (z y) = 0), (II) ( x, y X) ((x (x y)) y = 0), (III) ( x X) (x x = 0), (IV) ( x, y X) (x y = 0, y x = 0 x = y). If a BCI-algebra X satisfies the following identity: (V) ( x X) (0 x = 0), then X is called a BCK-algebra. We can define a partial ordering on X by x y if only if x y = 0. Any BCK/BCI-algebra X satisfies the following conditions: ( x X) (x 0 = x), (4) ( x, y, z X) (x x z y z, z y z x), (5) ( x, y, z X) ((x y) z = (x z) y), (6) ( x, y, z X) ((x z) (y z) x y). (7) A nonempty subset S of a BCK/BCI-algebra X is called a subalgebra of X if x y S for all x, y S. A subset I of a BCK/BCI-algebra X is called an ideal of X if 0 I the following condition is valid. ( x, y X) (x y I, y I x I). (8) We refer the reader to the books [14,15] the paper [16] for further information regarding BCK/BCI-algebras. 3. Cubic Interval-Valued Intuitionistic Fuzzy Sets Definition 1. Let X be a nonempty set. By a cubic interval-valued intuitionistic fuzzy set (briefly, cubic IVIF set) in X we mean a structure A = { x, µ(x), Ã(x) x X} in which µ is a fuzzy set in X Ã is an interval-valued intuitionistic fuzzy set in X. A cubic IVIF set A = { x, µ(x), Ã(x) x X} is simply denoted by A = µ, Ã. Let A = µ, Ã be a cubic IVIF set in a nonempty set X. Given ε [0, 1] ([s α, t α ], [s β, t β ]) D[0, 1] D[0, 1], we consider the sets µ[ε] := {x X µ(x) ε}, α[ã][s α, t α ] := {x X α[ã](x) [s α, t α ]}, β[ã][s β, t β ] := {x X β[ã](x) [s β, t β ]}. Definition 2. Let X be a nonempty set. A cubic IVIF set A = µ, Ã is said to be α-internal if µ(x) α[ã](x) for all x X, β-internal if µ(x) β[ã](x) for all x X, internal if it is both α-internal β-internal. α-external if µ(x) / (α[ã] (x), α[ã] + (x)) for all x X. β-external if µ(x) / (β[ã] (x), β[ã] + (x)) for all x X. external if it is both α-external β-external. It is clear that if A = µ, Ã is a cubic IVIF set in X with α[ã](x) β[ã](x) = for some x X, then A = µ, Ã cannot be internal.

4 Axioms 2018, 7, 7 4 of 17 Example 1. Let A = µ, à be a cubic IVIF set in I in which Ã(x) = ([0.3, 0.5], [0.1, 0.4]) µ(x) = k for all x X. (1) If k (0.4, 0.5], then A = µ, à is α-internal which is not β-internal. (2) If k [0.1, 0.3), then A = µ, à is β-internal which is not α-internal. (3) If k [0.3, 0.4], then A = µ, à is internal. (4) If either k 0.5 or k 0.1, then A = µ, à is external. (5) If either k 0.5 or k 0.3, then A = µ, à is α-external but may not be β-external. (6) If either k 0.4 or k 0.1, then A = µ, à is β-external but may not be α-external. Proposition 1. Let A = µ, à be a cubic IVIF set in a nonempty set X in which β[ã] (x) α[ã] (x) < α[ã] + (x) β[ã] + (x) for all x X. If A = µ, à is α-internal, then it is β-internal so internal. Also, if A = µ, à is β-external, then it is α-external so external. Proof. Straightforward. Proposition 2. If A = µ, à is a cubic IVIF set in X which is not external, then there exist x, y X such that µ(x) α[ã](x) or µ(y) β[ã](y). Proof. Straightforward. Theorem 1. Let A = µ, à be a cubic IVIF set in X in which the right end point of α[ã](x) (or β[ã](x)) is equal to the left end point of β[ã](x) (or α[ã](x)) for all x X. If we define µ by ( x X) ( µ(x) = α[ã] + (x) = β[ã] (x) or µ(x) = α[ã] (x) = β[ã] + (x) ), then A = µ, à is external. Proof. Straightforward. For any cubic IVIF set A = µ, à in X, let U α (Ã) := {α[ã] + (x) x X}, L α (Ã) := {α[ã] (x) x X}, U β (Ã) := {β[ã] + (x) x X}, L β (Ã) := {β[ã] (x) x X}. Theorem 2. Let A = µ, à be a cubic IVIF set in X. If A = µ, à is both α-internal α-external (resp., both β-internal β-external), then µ(x) U α (Ã) L α (Ã) (resp., µ(x) U β (Ã) L β (Ã)) for all x X. Proof. Assume that A = µ, à is both α-internal α-external. Then µ(x) α[ã](x) µ(x) / ( α[ã] (x), α[ã] + (x) ) for all x X. It follows that µ(x) = α[ã] (x) or µ(x) = α[ã] + (x), that is, µ(x) L α (Ã) or µ(x) U α (Ã). Hence µ(x) U α (Ã) L α (Ã) for all x X. Similarly, if A = µ, à is both β-internal β-external, then µ(x) U α (Ã) L α (Ã) for all x X. Given an IVIF set à = (α[ã], β[ã]) in X, the complement of à is denoted by à c is defined as follows: where à c : X D[0, 1] D[0, 1], x α[ã c ], β[ã c ] (9) α[ã c ] : X D[0, 1], x [α[ã c ] (x), α[ã c ] + (x)] (10)

5 Axioms 2018, 7, 7 5 of 17 β[ã c ] : X D[0, 1], x [β[ã c ] (x), β[ã c ] + (x)] (11) with α[ã c ] (x) = 1 α[ã] + (x), α[ã c ] + (x) = 1 α[ã] (x), β[ã c ] (x) = 1 β[ã] + (x), β[ã c ] + (x) = 1 β[ã] (x). Definition 3. For any cubic IVIF sets A = µ, à B = λ, B in X, we define (Equality) A = B à = B µ = λ, (12) (P-order) A P B à B µ λ, (13) (R-order) A R B à B µ λ, (14) where à B means that α[ã] α[ B] β[ã] β[ B], that is, ( α[ã] (x) α[ B] (x), α[ã] + (x) α[ B] + (x) ( x X) β[ã] (x) β[ B] (x), β[ã] + (x) β[ B] + (x) ). It is clear that the set of all cubic IVIF sets in X forms a poset under the P-order P the R-order R. For any à i = { x, α[ã i ](x), β[ã i ](x) } where i Λ, we define à i = { } x, α[ à i ](x), β[ à i ](x) x X (15) à i = { } x, α[ à i ](x), β[ à i ](x) x X (16) where α[ à i ](x) = rsupα[ã i ](x), β[ à i ](x) = rinf β[ã i ](x), α[ à i ](x) = rinf α[ã i ](x), β[ à i ](x) = rsupβ[ã i ](x). Definition 4. Given a family {A i = µ i, à i i Λ} of cubic IVIF sets in X, we define (1) P A i := µ i, à i (2) R A i := µ i, à i (3) P A i := µ i, à i (4) R A i := µ i, à i (P-union) (R-union) (P-intersection) (R-intersection) Proposition 3. For any cubic IVIF sets A = µ, Ã, B = λ, B, C = ν, C D = ρ, D in X, we have (1) A P B, A P C A P B P C. (2) A R B, A R C A R B R C. (3) A P C, B P C A P B P C. (4) A R C, B R C A R B R C. (5) A P B, C P D A P C P B P D, A P C P B P D. (6) A R B, C R D A R C R B R D, A R C R B R D.

6 Axioms 2018, 7, 7 6 of 17 Proof. Straightforward. Theorem 3. Let A = µ, à be a cubic IVIF set in X. If A = µ, à is α-internal (resp., α-external), then so is the complement of A = µ, Ã. Proof. Assume that A = µ, à is α-internal. Then µ(x) α[ã](x), that is, α[ã] (x) µ(x) α[ã] + (x) for all x X. Thus 1 α[ã] + (x) 1 µ(x) 1 α[ã] (x), that is, µ c (x) α[ã c ](x) for all x X. Hence A c = µ c, à c is α-internal. If A = µ, à is α-external, then µ(x) / (α[ã] (x), α[ã] + (x)) for all x X, so µ(x) [0, α[ã] (x)] or µ(x) [α[ã] + (x), 1]. It follows that 1 α[ã] (x) 1 µ(x) 1 or 0 1 µ(x) 1 α[ã] + (x), so that Therefore A c = µ c, à c is α-external. µ c (x) = 1 µ(x) / (1 α[ã] + (x), 1 α[ã] (x)) = (α[ã c ] (x), α[ã c ] + (x)) Theorem 4. Let A = µ, à be a cubic IVIF set in X. If A = µ, à is β-internal (resp., β-external), then so is the complement of A = µ, Ã. Proof. It is similar to the proof of Theorem 3. Corollary 1. If a cubic IVIF set A = µ, à in X is internal, then so is A c = µ c, à c. Theorem 5. If {A i = µ i, à i i Λ} is a family of α-internal cubic IVIF sets in X, then the P-union the P-intersection of {A i = µ i, à i i Λ} are also α-internal cubic IVIF sets in X. Proof. Since A i is α-internal, we have µ i (x) α[ã i ](x), that is, for all x X i Λ. It follows that α[ã i ] (x) µ i (x) α[ã i ] + (x) α[ à i ] (x) α[ à i ] (x) ( ) µ i (x) α[ à i ] + (x) ( ) µ i (x) α[ à i ] + (x). Therefore P A i P A i are α-internal cubic IVIF sets in X. The following example shows that the R-union R-intersection of α-internal cubic IVIF sets need not be α-internal. Example 2. Let A = µ, à B = λ, B be cubic IVIF sets in I = [0, 1] with Ã(x) = ([0.2, 0.45], [0.3, 0.5]) B(x) = ([0.6, 0.8], [0.1, 0.2]) for all x I. Then à B = (α[ã B], β[ã B]) à B = (α[ã B], β[ã B]), where α[ã B](x) = rmax{α[ã](x), β[ B](x)} = rmax{[0.2, 0.45], [0.6, 0.8]} = [0.6, 0.8]

7 Axioms 2018, 7, 7 7 of 17 β[ã B](x) = rmin{β[ã](x), β[ B](x)} = rmin{[0.3, 0.5], [0.1, 0.2]} = [0.1, 0.2] α[ã B](x) = rmin{α[ã](x), α[ B](x)} = rmin{[0.2, 0.45], [0.6, 0.8]} = [0.2, 0.45] β[ã B](x) = rmax{β[ã](x), β[ B](x)} = rmin{[0.3, 0.5], [0.1, 0.2]} = [0.3, 0.5]. If µ(x) = 0.4 λ(x) = 0.7 for all x I, then A B are α-internal. The R-union of A B is A R B = { x, µ(x), B(x) x I} it is not α-internal. The R-intersection of A B is which is not α-internal. A R B = { x, λ(x), Ã(x) x I} We provide a condition for the R-union of two α-internal cubic IVIF sets to be α-internal. Theorem 6. Let A = µ, Ã B = λ, B be cubic IVIF sets in X such that ( x X) ( max{α[ã] (x), α[ B] (x)} (µ λ)(x) ). (17) If A = µ, Ã B = λ, B are α-internal, then so is the R-union of A = µ, Ã B = λ, B. Proof. Assume that A = µ, Ã B = λ, B are α-internal cubic IVIF sets in X that satisfies the condition (17). Then α[ã] (x) µ(x) α[ã] + (x) α[ B] (x) λ(x) α[ B] + (x) for all x X, which imply that (µ λ)(x) α[ã B] + (x). It follows from (17) that for all x X. Therefore α[ã B] (x) = max{α[ã] (x), α[ B] (x)} (µ λ)(x) α[ã B] + (x) is an α-internal cubic IVIF set in X. A R B = { x, (µ λ)(x), (Ã B)(x) x X} We provide a condition for the R-intersection of two α-internal cubic IVIF sets to be α-internal. Theorem 7. Let A = µ, Ã B = λ, B be cubic IVIF sets in X such that ( x X) ( min{α[ã] + (x), α[ B] + (x)} (µ λ)(x) ). (18) If A = µ, Ã B = λ, B are α-internal, then so is the R-intersection of A = µ, Ã B = λ, B.

8 Axioms 2018, 7, 7 8 of 17 Proof. Let A = µ, Ã B = λ, B be α-internal cubic IVIF sets in X which satisfy the condition (18). Then α[ã] (x) µ(x) α[ã] + (x) α[ B] (x) λ(x) α[ B] + (x) for all x X. It follows from (18) that for all x X. Therefore α[ã B] (x) (µ λ)(x) min{α[ã] + (x), α[ B] + (x)} = α[ã B] + (x) is an α-internal cubic IVIF set in X. A R B = { x, (µ λ)(x), (Ã B)(x) x X} The following example shows that the P-union the P-intersection of α-external cubic IVIF sets need not be α-external. Example 3. Let A = µ, Ã B = λ, B be cubic IVIF sets in X which are given in Example 2. If µ(x) = 0.7 λ(x) = 0.4 for all x I, then A B are α-external. The P-union of A B is A P B = { x, µ(x), B(x) x I} it is not α-external. The P-intersection of A B is which is not α-external. A P B = { x, λ(x), Ã(x) x I} We consider conditions for the P-union P-intersection of two α-external cubic IVIF sets to be α-external. Theorem 8. Let A = µ, Ã B = λ, B be cubic IVIF sets in X such that {x X µ(x) α[ã] (x), λ(x) α[ B] + (x)} = (19) {x X µ(x) α[ã] + (x), λ(x) α[ B] (x)} =. (20) If A = µ, Ã B = λ, B are α-external, then so are the P-union P-intersection of A = µ, Ã B = λ, B. Proof. If A = µ, Ã B = λ, B are α-external, then µ(x) λ(x) / ( α[ B] (x), α[ B] + (x) ), that is, / ( α[ã] (x), α[ã] + (x) ) µ(x) α[ã] (x) or µ(x) α[ã] + (x) λ(x) α[ B] (x) or λ(x) α[ B] + (x) for all x X. It follows from conditions (19) (20) that ( x X) ( µ(x) α[ã] (x), λ(x) α[ B] (x) ) (21)

9 Axioms 2018, 7, 7 9 of 17 or The conditions (21) (22) induce respectively, for all x X. Hence for all x X, therefore ( x X) ( µ(x) α[ã] + (x), λ(x) α[ B] + (x) ). (22) (µ λ)(x) max{α[ã] (x), α[ B] (x)} (µ λ)(x) max{α[ã] + (x), α[ B] + (x)} (µ λ)(x) / ( α[ã B] (x), α[ã B] + (x) ) A P B = { x, (µ λ)(x), (à B)(x) x I} is an α-external cubic IVIF set in X. Also, (21) (22) imply that respectively, for all x X. Therefore is an α-external cubic IVIF set in X. (µ λ)(x) min{α[ã] (x), α[ B] (x)} (µ λ)(x) min{α[ã] + (x), α[ B] + (x)} A P B = { x, (µ λ)(x), (à B)(x) x I} The following example shows that the R-union R-intersection of α-external cubic IVIF sets in X may not be α-external. Example 4. Let A = µ, à B = λ, B be cubic IVIF sets in I = [0, 1] with Ã(x) = ([0.2, 0.4], [0.3, 0.5]) B(x) = ([0.3, 0.6], [0.2, 0.4]) for all x I. Then à B = (α[ã B], β[ã B]) à B = (α[ã B], β[ã B]), where α[ã B](x) = rmax{α[ã](x), α[ B](x)} = rmax{[0.2, 0.4], [0.3, 0.6]} = [0.3, 0.6] β[ã B](x) = rmin{β[ã](x), β[ B](x)} = rmin{[0.3, 0.5], [0.2, 0.4]} = [0.2, 0.4] α[ã B](x) = rmin{α[ã](x), α[ B](x)} = rmin{[0.2, 0.4], [0.3, 0.6]} = [0.2, 0.4] β[ã B](x) = rmax{β[ã](x), β[ B](x)} = rmin{[0.3, 0.5], [0.2, 0.4]} = [0.3, 0.5].

10 Axioms 2018, 7, 7 10 of 17 If µ(x) = 0.5 λ(x) = 0.7 for all x I, then A B are both α-external. The R-union of A B is A R B = { x, µ(x), B(x) x I} it is not α-external. If µ(x) = 0.1 λ(x) = 0.3 for all x I, then A B are α-external. The R-intersection of A B is which is not α-external. A R B = { x, λ(x), Ã(x) x I} We discuss conditions for the R-union R-intersection of two cubic IVIF sets to be α-external. Let A = µ, Ã B = λ, B be cubic IVIF sets in X such that ( x X) ( α[ã] (x) α[ B] (x) α[ã] + (x) α[ B] + (x) ). (23) If µ(x) α[ã] (x) for all x X, then (µ λ)(x) α[ã] (x) max{α[ã] (x), α[ B] (x)} so (µ λ)(x) / ( α[ã B] (x), α[ã B] + (x) ). If (µ λ)(x) α[ B] + (x) for all x X, then (µ λ)(x) α[ B] + (x) = max{α[ã] + (x), α[ B] + (x)} thus (µ λ)(x) / ( α[ã B] (x), α[ã B] + (x) ). If (µ λ)(x) α[ã] (x) for all x X, then (µ λ)(x) α[ã] (x) = min{α[ã] (x), α[ B] (x)} which implies that (µ λ)(x) / ( α[ã B] (x), α[ã B] + (x) ). If µ(x) α[ B] + (x) for all x X, then (µ λ)(x) µ(x) α[ B] + (x) min{α[ã] + (x), α[ B] + (x)} which induces (µ λ)(x) / ( α[ã B] (x), α[ã B] + (x) ). Therefore we have the following theorem. Theorem 9. Let A = µ, Ã B = λ, B be cubic IVIF sets in X satisfying the condition (23). If µ λ satisfies any one of the following conditions. ( x X) ( µ(x) α[ã] (x) ), (24) ( x X) ( (µ λ)(x) α[ B] + (x) ), (25) then the R-union of A = µ, Ã B = λ, B is α-external. following conditions. If µ λ satisfies any one of the then the R-intersection of A = µ, Ã B = λ, B is α-external. ( x X) ( (µ λ)(x) α[ã] (x) ), (26) ( x X) ( µ(x) α[ B] + (x) ), (27)

11 Axioms 2018, 7, 7 11 of 17 Let A = µ, Ã B = λ, B be cubic IVIF sets in X such that ( x X) ( α[ B] (x) α[ã] (x) α[ã] + (x) α[ B] + (x) ). (28) If µ(x) α[ B] (x) for all x X, then (µ λ)(x) α[ B] (x) max{α[ã] (x), α[ B] (x)} so (µ λ)(x) / ( α[ã B] (x), α[ã B] + (x) ). If (µ λ)(x) α[ B] + (x) for all x X, then (µ λ)(x) α[ B] + (x) = max{α[ã] + (x), α[ B] + (x)} thus (µ λ)(x) / ( α[ã B] (x), α[ã B] + (x) ). If {x X µ(x) α[ B] + (x)} = X, then (µ λ)(x) µ(x) α[ B] + (x) min{α[ã] + (x), α[ B] + (x)} which induces (µ λ)(x) / ( α[ã B] (x), α[ã B] + (x) ) for all x X. If {x X (µ λ)(x) α[ B] (x)} = X, then (µ λ)(x) α[ B] (x) = min{α[ã] (x), α[ B] (x)} so (µ λ)(x) / ( α[ã B] (x), α[ã B] + (x) ) for all x X. Therefore we have the following theorem. Theorem 10. Let A = µ, Ã B = λ, B be cubic IVIF sets in X satisfying the condition (28). If µ λ satisfies or {x X µ(x) α[ B] (x)} = X {x X (µ λ)(x) α[ B] + (x)} = X, then the R-union of A = µ, Ã B = λ, B is α-external. Also, if or {x X µ(x) α[ B] + (x)} = X {x X (µ λ)(x) α[ B] (x)} = X, then the R-intersection of A = µ, Ã B = λ, B is α-external. 4. Cubic IVIF Subalgebras Ideals In what follows, let X be a BCK/BCI-algebra unless otherwise specified. Definition 5. A cubic IVIF set A = µ, Ã in X is called a cubic IVIF subalgebra of X if the following conditions are valid. ( x, y X) (µ(x y) min{µ(x), µ(y)}), (29) ( ) α[ã](x y) rmax{α[ã](x), α[ã](y)} ( x, y X). (30) β[ã](x y) rmin{β[ã](x), β[ã](y)}

12 Axioms 2018, 7, 7 12 of 17 Example 5. Let X = {0, a, b, c} be a BCK-algebra with the Cayley table (Table 1). Table 1. Tabular representation of the binary operation. 0 a b c a a 0 0 a b b a 0 b c c c c 0 Define a cubic IVIF set A = µ, Ã in X by the tabular representation in Table 2. Table 2. Tabular representation of A = µ, Ã. X µ Ã = ( α[ã], β[ã] ) ([0.1, 0.4], [0.3, 0.5]) a 0.3 ([0.2, 0.4], [0.2, 0.4]) b 0.6 ([0.2, 0.5], [0.1, 0.3]) c 0.5 ([0.3, 0.5], [0.2, 0.5]) It is routine to verify that A = µ, Ã is a cubic IVIF subalgebra of X. Proposition 4. If A = µ, Ã is a cubic IVIF subalgebra of X, then µ(0) µ(x), α[ã](0) α[ã](x) β[ã](0) β[ã](x) for all x X. Proof. Since x x = 0 for all x X, we have This completes the proof. µ(0) = µ(x x) min{µ(x), µ(x)} = µ(x), α[ã](0) = α[ã](x x) rmax{α[ã](x), α[ã](x)} = α[ã](x), β[ã](0) = β[ã](x x) rmin{β[ã](x), β[ã](x)} = β[ã](x). Proposition 5. If A = µ, Ã is a cubic IVIF subalgebra of a BCI-algebra X, then µ(0 x) µ(x), α[ã](0 x) α[ã](x) β[ã](0 x) β[ã](x) for all x X. Proof. It follows from Definition 5 Proposition 4. Proposition 6. For a cubic IVIF subalgebra A = µ, Ã of X, if there exists a sequence {x n } in X such that lim µ(x n) = 1, lim α[ã](x n ) = [0, 0] lim β[ã](x n ) = [1, 1], then µ(0) = 1, α[ã](0) = [0, 0] n n n β[ã](0) = [1, 1]. Proof. Using Proposition 4, we have µ(0) µ(x n ), α[ã](0) α[ã](x n ) β[ã](0) β[ã](x n ). It follows from hypothesis that 1 µ(0) lim n µ(x n ) = 1, [0, 0] α[ã](0) lim n α[ã](x n ) = [0, 0], [1, 1] β[ã](0) lim n β[ã](x n ) = [1, 1], Hence µ(0) = 1, α[ã](0) = [0, 0] β[ã](0) = [1, 1].

13 Axioms 2018, 7, 7 13 of 17 Theorem 11. If A = µ, Ã is a cubic IVIF subalgebra of X, then the sets µ[ε], α[ã][s, t] β[ã][s, t], are subalgebras of X for all ε [0, 1] [s, t] D[0, 1]. Proof. For any ε [0, 1] [s, t] D[0, 1], let x, y X be such that x, y µ[ε] α[ã][s, t] β[ã][s, t]. Then µ(x) ε, µ(y) ε, α[ã](x) [s, t], α[ã](y) [s, t], β[ã](x) [s, t] β[ã](y) [s, t]. It follows that µ(x y) min{µ(x), µ(y)} min{ε, ε} = ε, α[ã](x y) rmax{α[ã](x), α[ã](y)} rmax{[s, t], [s, t]} = [s, t], β[ã](x y) rmin{β[ã](x), β[ã](y)} rmin{[s, t], [s, t]} = [s, t], that is, x y µ[ε], x y α[ã][s, t] x y β[ã][s, t]. Therefore µ[ε], α[ã][s, t] β[ã][s, t] are subalgebras of X for all ε [0, 1] [s, t] D[0, 1]. Corollary 2. If A = µ, Ã is a cubic IVIF subalgebra of X, then µ[ε] α[ã][s, t] β[ã][s, t] is a subalgebra of X for all ε [0, 1] [s, t] D[0, 1]. The following example shows that the converse of Corollary 2 is not true in general. Example 6. Let X = {0, 1, 2, 3, 4} be a set with the Cayley table (Table 3). Table 3. Tabular representation of the binary operation Then X is a BCK-algebra (see [15]). Define a cubic IVIF set A = µ, Ã in X by the tabular representation in Table 4. Table 4. Tabular representation of A = µ, Ã. X µ Ã = ( α[ã], β[ã] ) ([0.1, 0.3], [0.3, 0.6]) ([0.2, 0.4], [0.2, 0.4]) ([0.2, 0.5], [0.1, 0.3]) ([0.3, 0.5], [0.2, 0.5]) ([0.3, 0.5], [0.2, 0.5]) It is routine to verify that µ[ε] α[ã][s, t] β[ã][s, t] is a subalgebra of X for all ε [0, 1] [s, t] D[0, 1]. But, A = µ, Ã is not a cubic IVIF subalgebra of X since µ(2 3) = µ(1) = 0.3 < 0.5 = min{µ(2), µ(3)}. We provide conditions for a cubic IVIF set A = µ, Ã in X to be a cubic IVIF subalgebra of X. Theorem 12. Let A = µ, Ã be a cubic IVIF set in X such that µ[ε], α[ã][s α, t α ] β[ã][s β, t β ] are subalgebras of X for all ε [0, 1], ([s α, t α ], [s β, t β ]) D[0, 1] D[0, 1]. Then A = µ, Ã is a cubic IVIF subalgebra of X.

14 Axioms 2018, 7, 7 14 of 17 Proof. For any x, y X, putting min{µ(x), µ(y)} := ε rmax{α[ã](x), α[ã](y)} := [s α, t α ], rmin{β[ã](x), β[ã](y)} := [s β, t β ], imply that µ(x) ε, µ(y) ε, α[ã](x) [s α, t α ], α[ã](y) [s α, t α ], β[ã](x) [s β, t β ], β[ã](y) [s β, t β ]. Hence x µ[ε], y µ[ε], x α[ã][s α, t α ], y α[ã][s α, t α ], x β[ã][s β, t β ], y β[ã][s β, t β ]. It follows from hypothesis that x y µ[ε], x y α[ã][s α, t α ], x y β[ã][s β, t β ], so that µ(x y) ε = min{µ(x), µ(y)} α[ã](x y) [s α, t α ] = rmax{α[ã](x), α[ã](y)}, β[ã](x y) [s β, t β ] = rmin{β[ã](x), β[ã](y)}. Therefore A = µ, Ã is a cubic IVIF subalgebra of X. The following theorem gives us a way to establish a new cubic IVIF subalgebra from old one in BCI-algebras. Theorem 13. Given a cubic IVIF subalgebra A = µ, Ã of a BCI-algebra X, let A = µ, A be a cubic IVIF set in X defined by µ (x) = µ(0 x), α[ã] (x) = α[ã](0 x) β[ã] (x) = β[ã](0 x) for all x X. Then A = µ, A is a cubic IVIF subalgebra of X. Proof. Since 0 (x y) = (0 x) (0 y) for all x, y X, it follows that µ (x y) = µ(0 (x y)) = µ((0 x) (0 y)) min{µ(0 x), µ(0 y)} = min{µ (x), µ (y)}, α[ã] (x y) = α[ã](0 (x y)) = α[ã]((0 x) (0 y)) rmax{α[ã](0 x), α[ã](0 y)} = rmax{α[ã] (x), α[ã] (y)}, β[ã] (x y) = β[ã](0 (x y)) = β[ã]((0 x) (0 y)) rmin{β[ã](0 x), β[ã](0 y)} = rmin{β[ã] (x), β[ã] (y)} for all x, y X. Therefore A = µ, A is a cubic IVIF subalgebra of X.

15 Axioms 2018, 7, 7 15 of 17 Definition 6. A cubic IVIF set A = µ, Ã is called a cubic IVIF ideal of X if the following conditions are valid. ( x X) (µ(0) µ(x)), (31) ( x X) ( α[ã](0) α[ã](x), β[ã](0) β[ã](x) ) (32) ( x, y X) (µ(x) min{µ(x y), µ(y)}) (33) ( ) α[ã](x) rmax{α[ã](x y), α[ã](y)} ( x, y X) (34) β[ã](x) rmin{β[ã](x y), β[ã](y)} Example 7. Let X = {0, a, 1, 2, 3} be a BCI-algebra with the Cayley table in Table 5. Table 5. Tabular representation of the binary operation. 0 a a a Define a cubic IVIF set A = µ, Ã in X by Table 6. Table 6. Tabular representation of A = µ, Ã. X µ Ã = ( α[ã], β[ã] ) ([0.1, 0.3], [0.5, 0.6]) a 0.7 ([0.2, 0.4], [0.4, 0.5]) ([0.3, 0.6], [0.2, 0.3]) ([0.3, 0.6], [0.3, 0.4]) ([0.3, 0.6], [0.2, 0.3]) Then A = µ, Ã is a cubic IVIF ideal of X. Proposition 7. Every cubic IVIF ideal A = µ, Ã of X satisfies: µ(x) min{µ(y), µ(z)}. α[ã](x) rmax{α[ã](y), α[ã](z)}, β[ã](x) rmin{β[ã](y), β[ã](z)}, (35) for all x, y, z X with x y z. Proof. Let x, y, z X be such that x y z. Then (x y) z = 0, so µ(x y) min{µ((x y) z), µ(z)} = min{µ(0), µ(z)} = µ(z), α[ã](x y) rmax{α[ã]((x y) z), α[ã](z)} = rmax{α[ã](0), α[ã](z)} = α[ã](z), β[ã](x y) rmin{β[ã]((x y) z), β[ã](z)} = rmin{β[ã](0), β[ã](z)} = β[ã](z). It follows that µ(x) min{µ(x y), µ(y)} min{µ(y), µ(z)}, α[ã](x) rmax{α[ã](x y), α[ã](y)} rmax{α[ã](y), α[ã](z)}, β[ã](x) rmin{β[ã](x y), β[ã](y)} rmin{β[ã](y), β[ã](z)}.

16 Axioms 2018, 7, 7 16 of 17 This completes the proof. We provide conditions for a cubic IVIF set to be a cubic IVIF ideal. Theorem 14. Let A = µ, Ã be a cubic IVIF set in X in which conditions (31) (32) are valid. If A = µ, Ã satisfies the condition (35) for all x, y, z X with x y z, then A = µ, Ã is a cubic IVIF ideal of X. Proof. Since x (x y) y for all x, y X, it follows from (35) that µ(x) min{µ(x y), µ(y)}, α[ã](x) rmax{α[ã](x y), α[ã](y)}, β[ã](x) rmin{β[ã](x y), β[ã](y)}, Therefore A = µ, Ã is a cubic IVIF ideal of X. Lemma 1. Every cubic IVIF ideal A = µ, Ã in X satisfies the following condition. ( x, y X) ( x y µ(x) µ(y), α[ã](x) α[ã](y), β[ã](x) β[ã](y) ). (36) Proof. Assume that x y for all x, y X. Then x y = 0, so µ(x) min{µ(x y), µ(y)} = min{µ(0), µ(y)} = µ(y), α[ã](x) rmax{α[ã](x y), α[ã](y)} = rmax{α[ã](0), α[ã](y)} = α[ã](y), β[ã](x) rmin{β[ã](x y), β[ã](y)} = rmin{β[ã](0), β[ã](y)} = β[ã](y) by (31) (34). Theorem 15. In a BCK-algebra X, every cubic IVIF ideal is a cubic IVIF subalgebra. Proof. Let A = µ, Ã be a cubic IVIF ideal of a BCK-algebra X. Since x y x for all x, y X, we have µ(x y) µ(x), α[ã](x y) α[ã](x), β[ã](x y) β[ã](x) by Lemma 1. It follows from (33) (34) that µ(x y) µ(x) min{µ(x y), µ(y)} min{µ(x), µ(y)}, α[ã](x y) α[ã](x) rmax{α[ã](x y), α[ã](y)} rmax{α[ã](x), α[ã](y)}, β[ã](x y) β[ã](x) rmin{β[ã](x y), β[ã](y)} rmin{β[ã](x), β[ã](y)}. Therefore A = µ, Ã is a cubic IVIF subalgebra of X. The converse of Theorem 15 is not true in general as seen in the following example. Example 8. The cubic IVIF subalgebra A = µ, Ã in Example 5 is not a cubic IVIF ideal of X since µ(a) = 0.3 < 0.6 = min{µ(a b), µ(b)}. We establish a characterization of a cubic IVIF ideal in a BCK-algebra. Theorem 16. For a cubic IVIF set A = µ, Ã in a BCK-algebra X, the following assertions are equivalent. (1) A = µ, Ã is a cubic IVIF ideal of X. (2) A = µ, Ã is a cubic IVIF subalgebra of X satisfying (35) for all x, y, z X with x y z.

17 Axioms 2018, 7, 7 17 of 17 Proof. (1) (2). It follows from Proposition 7 Theorem 15. (2) (1). It is by Proposition 4 Theorem Conclusions We have introduced a cubic interval-valued intuitionistic fuzzy set which is an extension of a cubic set, have applied it to BCK/BCI-algebra. We have investigated P-union, P-intersection, R-union R-intersection of α-internal α-external cubic IVIF sets. We have defined cubic IVIF subalgebra ideal in BCK/BCI-algebra, have investigated related properties. We have considered relations between cubic IVIF subalgebra cubic IVIF ideal. We have discussed characterizations of cubic IVIF subalgebra cubic IVIF ideal. In the consecutive research, we will discuss P-union, P-intersection, R-union R-intersection of β-internal β-external cubic IVIF sets. We will apply this notion to several kind of ideals in BCK/BCI-algebras, for example, (positive) implicative ideal, commutative ideal, associative ideal, q-ideal, etc. We will also apply this notion to several algebraic substructures in various algebraic structures, for example, MV-algebra, BL-algebra, R 0 -algebra, residuated lattice, MTL-algebra, semigroup, semiring, near-ring, etc. We will try to study this notion in the neutrosophic environment. Acknowledgments: The authors thank the anonymous reviewers for their valuable comments suggestions. Author Contributions: Young Bae Jun initiated the main idea of the work wrote the paper. Seok-Zun Song Young Bae Jun conceived designed the new definitions results. Seok-Zun Song Seon Jeong Kim performed finding examples checking contents. All authors have read approved the final manuscript for submission. Conflicts of Interest: The authors declare no conflict of interest. References 1. Jun, Y.B.; Kim, C.S.; Yang, K.O. Cubic sets. Ann. Fuzzy Math. Inform. 2012, 4, Ahn, S.S.; Kim, Y.H.; Ko, J.M. Cubic subalgebras filters of CI-algebras. Honam Math. J. 2014, 36, Akram, M.; Yaqoob, N.; Gulistan, M. Cubic KU-subalgebras. Int. J. Pure Appl. Math. 2013, 89, Jun, Y.B.; Khan, A. Cubic ideals in semigroups. Honam Math. J. 2013, 35, Jun, Y.B.; Kim, C.S.; Kang, J.G. Cubic q-ideals of BCI-algebras. Ann. Fuzzy Math. Inform. 2011, 1, Jun, Y.B.; Lee, K.J. Closed cubic ideals cubic -subalgebras in BCK/BCI-algebras. Appl. Math. Sci. 2010, 4, Jun, Y.B.; Lee, K.J.; Kang, M.S. Cubic structures applied to ideals of BCI-algebras. Comput. Math. Appl. 2011, 62, Khan, M.; Jun, Y.B.; Gulistan, M.; Yaqoob, N. The generalized version of Jun s cubic sets in semigroups. J. Intell. Fuzzy Syst. 2015, 28, Muhiuddin, G.; Al-roqi, A.M. Cubic soft sets with applications in BCK/BCI-algebras. Ann. Fuzzy Math. Inform. 2014, 8, Senapati, T.; Kim, C.S.; Bhowmik, M.; Pal, M. Cubic subalgebras cubic closed ideals of B-algebras. Fuzzy Inf. Eng. 2015, 7, Yaqoob, N.; Mostafa, S.M.; Ansari, M.A. On cubic KU-ideals of KU-algebras. ISRN Algebra 2013, doi: /2013/ Atanassov, K.; Gargov, G. Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 1989, 31, Atanassov, K. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20, Huang, Y. BCI-Algebra; Science Press: Beijing, China, Meng, J.; Jun, Y.B. BCK-Algebras; Kyungmoon Sa Co.: Seoul, Korea, Iséki, K.; Tanaka, S. An introduction to the theory of BCK-algebras. Math. Jpn. 1978, 23, c 2018 by the authors. Licensee MDPI, Basel, Switzerl. This article is an open access article distributed under the terms conditions of the Creative Commons Attribution (CC BY) license (